Decisions, decisions, decisions in an uncertain environment

Decision-makers abhor uncertainty, and it is certainly true that the less there is of it the better. However, recognizing that uncertainty is part of the equation, particularly for deciding on environmental policy, is a prerequisite for making wise decisions. Even making no decision is a decision that has consequences, and using the presence of uncertainty as the reason for failing to act is a poor excuse. Statistical science is the science of uncertainty, and it should play a critical role in the decision-making process. This opinion piece focuses on the summit of the knowledge pyramid that starts from data and rises in steps from data to information, from information to knowledge, and finally from knowledge to decisions. Enormous advances have been made in the last 100 years ascending the pyramid, with deviations that have followed different routes. There has generally been a healthy supply of uncertainty quantification along the way but, in a rush to the top, where the decisions are made, uncertainty is often left behind. In my opinion, statistical science needs to be much more pro-active in evolving classical decision theory into a relevant and practical area of decision applications. This article follows several threads, building on the decision-theoretic foundations of loss functions and Bayesian uncertainty.

Making a decision under almost any circumstance comes with uncertainty. For example, a country’s investment in carbon capture and storage (CCS) comes with only a degree of certainty that the technology can be applied at scale. Australia’s current commitment to “net zero emissions by 2050” assumes that CCS, along with other technologies not even conceived of yet, will contribute substantially to reducing Australia’s greenhouse gas emissions. In the Australian Government’s 2021 plan, there is a prediction that “… future technology breakthroughs will reduce emissions by … 15 per cent by 2050” (Commonwealth of Australia, 2021). The remaining 85% appears to be less uncertain in the Australian Government’s technology-driven plan: 40% comes from driving down technology costs and accelerating their deployment at scale (Technology Investment Roadmap, 2021); 20% comes from emissions reductions already made since 2005; 15% comes from global technology trends; and 10% comes from high-integrity offsets. A decision was made by the Australian Government in 2021 to rule out taxes and legislative mechanisms as part of achieving net zero carbon emissions by 2050, in line with the ruling Liberal-National Party (LNP) coalition’s conservative agenda. (In the federal election of May 2022, the LNP were defeated by the Australian Labor Party, who promised to accelerate the passage to net zero carbon emissions – new decisions by the new Australian Government are forthcoming.)

Policy-makers make decisions based on data, modeling (depending on assumptions), predictions (from the modeling), and expert opinions (sometimes colored by the politics of the day). Where is there not uncertainty in this process, yet how often is it made transparent? Even the emissions reductions of 20% claimed between 2005 and 2021 in the Australian Government’s plan is based on industry and state-government self-reporting; deeper in the plan, a figure to one decimal place of 20.8% is given, indicating high certainty but based on figures whose uncertainty has almost certainly (!) not been quantified.

Statistical science is the science of uncertainty, which includes its quantification, its estimation, and its consequences for inference in scientific models and the decisions that follow. Statistical scientists have almost always focused on how uncertainty and variability affect inference on key parameters and (summaries of) scientific processes; however, much less of their literature is devoted to the effect of uncertainty on decisions. Sometimes decision-makers see uncertainty as a reason for lack of action, but that too is a decision, and it has consequences. Sometimes the uncertainty of inference on an important environmental parameter is provided but ignored by the decision-maker, and only the estimate remains, to be seen as precise as the number of digits quoted.

Science (environmental, medical, social, statistical, etc.) in its service to society should be better integrated into the peak of the knowledge pyramid, where decisions are made: At the base of the pyramid are data; at the next level up is information obtained by exploring the data for structure; the information is then converted into knowledge at the next level by modeling the uncertainty/variability and inferring etiology; and finally, at the peak, decisions are made. Uncertainty quantification should be involved at all levels, and the consequences of ignoring it in environmental problems can be serious. For example, deciding to evacuate neighbourhoods and townships faced with possible flooding is based on meteorological and hydrological forecasts. Uncertainty of both forecasts should jointly inform evacuation decisions by government agencies; communities cannot be kept safe “on average."

In the rest of this article, I shall discuss decision-making in the presence of uncertainty, opining that its future is in addressing the complex decisions that arise in uncertain environments. Section 2 reviews several approaches that have been taken in this area. Section 3 makes the case that decision theory and its applications have much to offer decision-makers, with its quantification of uncertainty through a posterior probability distribution and its quantification of the consequences of possible actions through a loss function. Several classes of loss functions are presented as alternatives to squared-error loss, including asymmetric loss functions that are key for making environmental decisions. Section 4 gives an example of how one such loss function can be calibrated. Finally, Section 5 gives concluding remarks and discusses new research directions.

Before committing effort and resources to making a decision, ask first “What is the question that the decision will answer?” In the example given in Section 1, the question is, “How will Australia achieve net zero carbon emissions by 2050?” It took about five years from the time of the COP21 Paris Agreement for the Australian Government to seriously address that question. The best questions focus the decision-maker to provide the “best” answers/decisions. Providing several “good” but sub-optimal answers is not all that bad, but lack of transparency through a “good-better-best” criterion is.

In that spirit, I am arguing for clarity of assumptions, a statement of the criteria and a common numéraire against which all competing decisions are evaluated, an error budget that sets out sources of uncertainty, a statistical model that quantifies the uncertainties, and inferences that incorporate uncertainty quantification. Not just any model will do and, guided by the error budget, decision-making in the presence of uncertainty is in my opinion best served with a well fitted Bayesian hierarchical model (BHM). Complexity in underlying processes can often be captured by a computer model (e.g., Santner et al., 2010) or what has been referred to as a “digital twin.” When building a BHM, the digital twin or some approximation to it could be used as the first moment of the hidden process.

Complexity is often the case in the environmental sciences, where there are many questions asked by many stakeholders. Taking a BHM approach means that all the answers can be found somewhere in a (high-dimensional) posterior distribution. This article addresses, inter alia, how one might organize the knowledge contained in the posterior distribution in order to make an optimal decision.

Section 3 presents a decision-theoretic approach to inference, with an emphasis on prediction of unknown random quantities, rather than on estimation of unknown parameters. A relatively new approach to making the difficult decisions one sees in environmental applications is VOI (Value of Information), which aims to understand what is gained by taking additional types and amounts of data (e.g., Eidsvik et al., 2015). Suppose “value” $V$ is measured by a particular forecast’s accuracy (economic, meteorological, epidemiological, etc.), but there is a suggestion that an extra data set might be incorporated to improve the accuracy. Then,

Of course, a threshold will need to be specified to decide whether the additional data adds enough value. In some instances, the extra data set is costly and requires a cost-of-sampling function to offset the data’s added value. As discussed in Section 5, incorporating the cost of data is an important part of sampling design; not all sensors are cheap, and “found” data can be of variable quality and value.

When the measurement error of the data is assumed to be zero, the information is deemed “perfect” and VOPI (Value of Perfect Information) is an aspirational VOI value: Then a proposed data set’s VOI can be compared to the corresponding VOPI (e.g., Lark et al., 2022). Use of VOI to make decisions is a relatively new approach, and practical issues are still being investigated.

Consider the basic problem of deciding between competing models, $M_{1},...,M_{m}$. One classical approach is to use Bayes factors (e.g., Berger, 1985). Define $\pi_{k}\equiv\mathrm{Pr}(M_{k})$, the prior probability that $M_{k}$ is the correct model, and assume that one of the $k=1,...,m$ models is correct. Informed by data $z$, the posterior probability that $M_{k}$ is the correct model is

Then $M_{k}$ could be compared to any of the other models, here $M_{j}$, by considering the Bayes factor,

and one could choose the optimal model, $\delta_{BAF}^{*}(z)\equiv\mathrm{argmax}\left\{p(k\mid z)/\pi_{k}:k=1,...,m\right\}$.

However, choosing the wrong model has differential consequences, depending on what is the correct model and which model was chosen. Another classical criterion is the expected posterior loss. Suppose those consequences are quantified by a decision table, which is an $m\times m$ matrix of losses $(L_{jk})$ where, for $j,k=1,...,m$, $L_{jk}$ is the loss incurred by choosing model $M_{j}$ when $M_{k}$ is the correct model, and $L_{jj}=0$. A decision to choose model $M_{j}$ has expected posterior loss,

Then, minimizing the $E_{PL}$ criterion (1) over $j\in\{1,...,m\}$ results in an optimal decision, which is to choose the model, $\delta_{EPL}^{*}(z)\equiv\mathrm{argmin}\left\{E_{PL}(j\mid z):j=1,...,m\right\}$.

These two decisions, $\delta_{BAF}^{*}$ and $\delta_{EPL}^{*}$, are only indirectly related, as can be seen if it is assumed that $L_{jk}$ is the indicator function, $I(j\neq k)$, sometimes known as the $0$-$1$ loss function. Then $E_{PL}(j\mid z)=1-p(j\mid z);j=1,...,m$, and hence $\delta_{EPL}^{*}(z)=\mathrm{argmax}\left\{p(j\mid z):j=1,...,m\right\}$. Therefore, under a uniform prior (i.e., $\pi_{j}=1/m$ for $j=1,...,m$), $\delta_{BAF}^{*}\equiv\delta_{EPL}^{*}$; otherwise, the two optimal decisions are different. This most basic of decision problems, to decide between models $M_{1},...,M_{m}$, was chosen to illustrate that different criteria may lead to different optimal decisions. Since the method of Bayes Factors optimizes the (frequentist) likelihood, my preference is for a method directly based on the (Bayesian) posterior distribution, such as the expected posterior loss given by (1). This is reinforced by the conclusions of Kass and Raftery (1995), that Bayes factors should be avoided when the prior is not uniform.

Decisions come with consequences (losses), and it is important to capture loss along with the uncertainty. The essential property of a loss function is that it be bounded from below; hence, without loss of generality, it can be assumed to be non-negative. If one is to use expected posterior loss as a criterion, then it must also be assumed that its expectation with respect to the posterior probability measure exists (an assumption that involves the behavior of both loss and uncertainty). Uncertainties expressed through a BHM are traditionally made up of parametric conditional probabilities. A different emphasis is given in the book by Smith (2010), who features the Bayes networks that support a BHM but devotes less to the quantification of its conditional probabilities.

We shall see in the next section that the expected posterior loss is a way to account for the high-impact, low-probability scenarios seen in the presence of extreme environmental threats. In my opinion, while statistical science has devoted much of its effort to the construction and properties of distribution functions for inference, it has spent far less effort on developing loss functions for decisions. Both are needed!

Uncertainty quantification based on conditional-probability models offers a framework for optimal decision-making. In particular, hierarchical statistical modeling effectively separates out the data from the latent process and the parameters. In this article, I focus principally on decisions about (functions of) random process(es). Decision theory provides a way to determine an optimal predictor $\delta^{*}(z)$ of the random quantity $Y$ based on data $z$ in the presence of uncertainty in both $Y$ and $z$.

Very simplistically, consider a set of actions $\{a^{(1)},a^{(2)},...\}$ and a loss function $L$, where $L$ quantifies the “balance sheet” of costs and benefits that depend on both the true quantity $Y$ and an action $a$ that could be taken by the decision-maker; hence, write it as $L(a,Y)$. This allows action $a^{(1)}$ to be compared to action $a^{(2)}$ for each possible realization $y$ of $Y$. The action with the lower loss is preferred, but a uniform ordering for all $y$ may not be achievable. In what follows, a solution based on decision theory is presented. For the case where a multivariable decision is needed, Section 3.2 proposes an eigenanalysis that reduces the problem to making independent decisions in orthogonal eigenspaces. Sections 3.3 and 3.4 develop new loss functions of action $a$ and random variable $Y$ that give the decision-maker plenty of alternatives to the ubiquitous squared-error loss function, $L_{SEL}(a,Y)\equiv(a-Y)^{2}$.

In Section 3, it was shown that loss functions abound, to capture circumstances where losses are not symmetric (e.g., squared-error loss) functions of displacement. When making decisions in the face of extreme events, asymmetric loss can be used to represent the often smaller cost of mitigation versus a major cost of recovery.

In this section, the class of LINEX loss functions $\{L_{LNX}^{(\psi)}(a,Y):\psi\in(-\infty,\infty)\}$, given by (8), quantifies this with a scalar tuning parameter $\psi$, where $\psi<0$ recognizes that larger losses will be incurred from underprediction of $Y$ (e.g., $Y$ is a river’s crest-height after an extreme amount of rainfall has occurred in the river’s catchment). For $|\psi|$ small, a Taylor-series expansion results in (approximate) squared-error loss: $L_{LNX}^{(\psi)}\approx(\psi^{2}/2)(a-Y)^{2}\propto(a-Y)^{2}$. However, as $\psi\rightarrow-\infty$, the loss when $a<Y$ (underprediction) grows exponentially compared to the loss when $a>Y$ (overprediction). Overprediction of the crest-height of a major river might lead to mass-evacuation orders that later could be seen as an unnecessary nuisance, but the alternative could be a very costly loss of material possessions/animals/crops/life. The asymmetric LINEX loss function can be tuned to make more conservative flood-related decisions by making $\psi$ more negative, resulting in a biased but conservative prediction of the crest-height. In this section, a way to calibrate $\psi$ is proposed.

Decisions are made in an uncertain environment, where here the uncertainty is expressed probabilistically, specifically through the joint distribution of $Y$ and $z$. We saw in Section 3.3 that the optimal predictor of $Y$ obtained from minimizing (2) with the LINEX loss function given by (8) is, for $\psi\in(-\infty,\infty)$,

where the approximation relies on small $\sigma^{2}_{y\mid z}$ and is obtained using the delta method in terms of the conditional moments, $\mu_{Y\mid z}\equiv E(Y\mid z)$ and $\sigma^{2}_{Y\mid z}\equiv\text{var}(Y\mid z)$. A further approximation up to $O(\sigma^{2}_{Y\mid z})$ is provided by the Taylor-series expansion, $\log(1+x)=x+O(x^{2})$; then from (15),

For $\psi<0$, the approximation shows that $\delta_{LNX}^{*}(z)$ conservatively overpredicts in relation to the posterior mean, $\mu_{Y\mid z}=E(Y\mid z)$. (It was shown in Section 3.3 that overprediction occurs for $\psi<0$, no matter how large $\sigma^{2}_{Y\mid z}$ is.) It is now shown that the approximation allows $\psi$ to be chosen according to a back-of-the-envelope calculation: Based on the $0.97$ Gaussian quantile, 1.88, which is motivated by the discussion at the end of Section 3.3, we solve for $\psi$ in $(-\psi\sigma_{Y\mid z})/2=1.88$, giving $\psi=-3.76/\sigma_{Y\mid z}$. Note that the calibration could be improved if more were known about $p(Y\mid z)$, ideally its moment generating function. Unlike the quantile loss function, the LINEX loss function has all derivatives, including $\partial L_{LNX}/\partial a$. In general, it can be advantangeous in higher-dimensional problems to use loss functions that allow calculation of derivatives to arrive at an optimal decision in terms of $E_{PL}$ given by (2).

The previous sections have followed several threads of decision theory and demonstrated the importance of quantifying a decision $a$ in terms of a loss $L(a,Y)$, where $Y$ is the unobserved random quantity of interest. Data $z$ that are dependent on $Y$ are available, and the uncertainty in $Y$ and $z$ is expressed probabilistically through the joint distribution $f_{12}(y,z)$. The two essential components for decision theory and its applications are $L(a,y)$ and $f_{12}(y,z)$. Concentrating on $E_{PL}$ requires working with the posterior distribution, which is calculated via Bayes’ Theorem, $p(y\mid z)=f_{12}(y,z)/f_{2}(z)=f(z\mid y)\pi(y)/f_{2}(z)$, where recall that $f(z\mid y)$ is the “data model” or the “likelihood”, $\pi(y)$ is the “process model,” and $f_{2}(z)$ is the marginal distribution of $z$ (also called “the normalizing constant”).

There are a number of key conclusions to draw:

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  An optimal decision depends fundamentally on the criterion being optimized.

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  A decision comes with consequences (here expressed as losses), which depend on both the hidden process $Y$ and on the decision $\delta(z)$ about $Y$, where $z$ represents the data. Loss functions are plentiful.

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  The hidden process $Y$ is uncertain, and that uncertainty can be expressed according to a probability model $\pi(\cdot)$ for $Y$.

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  After obtaining data $z$, the process model $\pi(\cdot)$ is updated to the posterior distribution $p(\cdot\mid z)$ according to Bayes’ Theorem, and that posterior should be used in the decision-making process, along with a loss function $L$.

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  The same optimal decision may come from quite different optimality criteria (defined by quite different loss functions).

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  The optimized criterion should not be forgotten and needs to be evaluated along with the optimal decision.

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  The optimality criterion is usually the expected posterior loss, $E_{PL}$ (but it does not have to be).

Classical decision theory has concentrated on two types of “expected loss,” namely Expected Joint Loss ($E_{JL}$), sometimes referred to as Bayes risk or unconditional risk:

and Expected Posterior Loss ($E_{PL}$), sometimes referred to as conditional risk:

In my opinion, the word “risk” should not be used to describe $E_{JL}$ and $E_{PL}$; “risk” should be used in a way that many decision-makers and the general public use it, namely as a probability. What’s in a name? Everything! Traditionalists may not like it, but if one adopts, say, the abbreviations “$E_{PL}$” to replace “conditional risk” and “$E_{JL}$” to replace “unconditional risk,” it would avoid the profound ambiguity that currently exists where, in common parlance, “risk” often means “probability”, not “expected loss.”

The decision-theoretic approach optimizes $E_{JL}$ with respect to decision $\delta$ or, equivalently, it opimizes $E_{PL}$ with respect to $\delta(z)$. The two ingredients of $E_{PL}$ are combined first as a product, $L(\delta(z),y)\times p(y\mid z)$, which is a way of weighting the consequence of a decision about an event (no matter how catastrophic) with the risk/probability of that event (no matter how small), given the data $z$. Recall that these products are integrated over all possible events to form $E_{PL}$, which is then minimized with respect to $\delta(z)\in\mathcal{A}$, for each $z$, resulting in the optimal predictor $\delta^{*}(\cdot)$.

A possible future direction is to find other ways of combining $\{L(\delta(z),y):y\in\mathcal{Y}\}$ and $\{p(y\mid z):y\in\mathcal{Y}\}$. One way that I do not advocate is minimizing the maximum loss, resulting in the so-called minimax decision, $\delta_{MML}(z)\equiv\mathrm{argmin}_{\delta(z)}\{\max_{y\in\mathcal{Y}}{L(\delta(z),y)}\}$, since it is blind to the uncertainty of the environment $Y$ by assuming all values of $y\in\mathcal{Y}$ are equally likely. However, minimizing the maximum posterior loss,
$\delta_{MMP}(z)\equiv\mathrm{argmin}_{\delta(z)}\{\max_{y\in\mathcal{Y}}{L(\delta(z),y)\times p(y\mid z)}\}$, does combine a value of $y$ with its posterior probability but, instead of averaging, it takes the maximum.

Just as the minimax criterion considers only $L$ and ignores $p$, the reverse sometimes happens where decisions are made based only on the posterior distribution $p$. For example, a “yes/no” decision might be based on a threshold $\kappa$ where “yes” corresponds to $\mathrm{Pr}(Y>\kappa)\geq 0.5$. Some reverse engineering from Section 3.3 can be used to demonstrate that the implied loss function for the yes/no decision is an unweighted $0$-$1$ loss function applied to the probability model, $\pi_{\kappa}\equiv\mathrm{Pr}(Y>\kappa)$ and $1-\pi_{\kappa}=\mathrm{Pr}(Y\leq\kappa)$. So under $0$-$1$ loss, one decides “yes” if $p_{\kappa}(z)\equiv\mathrm{Pr}(Y>\kappa\mid z)\geq 1-p_{\kappa}(z)$, which is equivalent to $\mathrm{Pr}(Y>\kappa\mid z)\geq 0.5$. However, “yes” might be more costly than “no,” which can be captured with a loss function given by a decision table (described in Section 2).

In Sections 3.3 and 3.4, it was shown that much more is possible to capture asymmetric loss than simply using posterior quantiles. For example, consider the weighted 0-1 and quantile loss functions:

which are particularly useful in the spatial context. Now the risk maps $\{\delta_{QTL}^{(q)}(z;\mathbf{s}):\mathbf{s}\in D\subset\mathbb{R}^{d}\}$, for various posterior quantiles over a spatial domain of interest $D$ (contained in $d$-dimensional Euclidean space $\mathbb{R}^{d}$, say), allow identification of at-risk regions in $D$. However, they do not use information about catastrophic losses that would be incurred should a threshold $\kappa$ be exceeded. That could happen by letting the weight $w(Y)$ depend on $\kappa$.